Math 618
Zeta function
April 19, 1999
The zeta function
The goal of this exercise is to understand the definition of the zeta function.
∞
∞
X
X
π2
π4
1
1
and
. What
You have likely seen formulas like
=
=
n2
6
n4
90
n=1
n=1
happens if the power of n is replaced by some other number? The result is
Riemann’s zeta function:
∞
X
1
,
Re z > 1.
(1)
ζ(z) :=
nz
n=1
1. Since the complex power ab := eb log a is “multi-valued”, is the definition
of ζ(z) ambiguous?
2. Why does the definition (1) produce a holomorphic function?
It turns out that the ζ function can be continued analytically to C \ {1},
with a simple pole at 1. Our goal is to prove a little less.
∞
X
(−1)n
= (21−z − 1)ζ(z) when Re z > 1 by grouping the
z
n
n=1
terms in the absolutely convergent sum on the left-hand side according
to the parity of n.
3. Show that
∞
X
(−1)n
represents a holomorphic function
nz
when Re z > 0, deduce that ζ is a meromorphic function when Re z > 0
and that ζ has a simple pole at 1 with residue 1.
4. Assuming that the series
n=1
5. A generalization of the alternating series test states that sufficient con∞
X
(−1)n bn are that the sequence {bn }
ditions for convergence of a series
n=1
has bounded variation, meaning
∞
X
|bn+1 − bn | < ∞, and that bn → 0.
n=1
(An analogous statement holds for uniform convergence.) To complete the preceding argument, deduce from this convergence test that
∞
X
(−1)n
does represent a holomorphic function when Re z > 0.
z
n
n=1
Theory of Functions of a Complex Variable II
Dr. Boas